Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지

Metric and Spectral Geometric Means on Symmetric Cones

  • Lee, Hosoo (Department of Mathematics, Kyungpook National Unversity) ;
  • Lim, Yongdo (Department of Mathematics, Kyungpook National Unversity)
  • Received : 2006.10.24
  • Published : 2007.03.23
Download PDF
⟨ Previous Next ⟩

Abstract

In a development of efficient primal-dual interior-points algorithms for self-scaled convex programming problems, one of the important properties of such cones is the existence and uniqueness of "scaling points". In this paper through the identification of scaling points with the notion of "(metric) geometric means" on symmetric cones, we extend several well-known matrix inequalities (the classical L$\ddot{o}$wner-Heinz inequality, Ando inequality, Jensen inequality, Furuta inequality) to symmetric cones. We also develop a theory of spectral geometric means on symmetric cones which has recently appeared in matrix theory and in the linear monotone complementarity problem for domains associated to symmetric cones. We derive Nesterov-Todd inequality using the spectral property of spectral geometric means on symmetric cones.

Keywords

References

  1. T. Ando, Topics on operator inequalities, Lecture Note Hokkaido Univ., Sapporo 1978.
  2. T. Ando, On some operator inequalities, Math. Ann., 279(1987), 157-159. https://doi.org/10.1007/BF01456197
  3. T. Ando and F. Hiai, Log majorization and complementary Golden-Thompson type inequalities, Linear Algebra Appl., 197/198(1994), 113-131. https://doi.org/10.1016/0024-3795(94)90484-7
  4. H. Bae and Y. Lim, On some Finsler structures of symmetric cones, Forum Math., to appear.
  5. Ph. Bougerol, Kalman filtering with random coefficients and contractions, SIAM J. Control Optim., 31(1993), 942-959. https://doi.org/10.1137/0331041
  6. G. Corach, H. Porta and L. Recht, Geodesics and operator means in the space of positive operators, International J. of Math., 4(1993), 193-202. https://doi.org/10.1142/S0129167X9300011X
  7. J. Faraut and A. Koranyi, Analysis on symmetric cones, Clarendon Press, Oxford 1994.
  8. L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. and Appl. Math., 86(1997), 149-175. https://doi.org/10.1016/S0377-0427(97)00153-2
  9. L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1(1997), 331-357. https://doi.org/10.1023/A:1009701824047
  10. L. Faybusovich, Euclidean Jordan algebras and generalized affine-scaling vector fields, preprint.
  11. L. Faybusovich, A Jordan-algebraic approach to potential-reduction algorithms, Math. Z., 239(2002), 117-129. https://doi.org/10.1007/s002090100286
  12. M. Fiedler and V. Ptak, A new positive definite geometric mean of two positive definite matrices, Linear Algebra Appl., 251(1997), 1-20. https://doi.org/10.1016/0024-3795(95)00540-4
  13. T. Furuta, A ${\geq}$ B ${\geq}$ 0 assures $(B^{r}\;A^{p}\;B^{r})^{1/q}$ ${\geq}$ $B^{(p+2r)/q}$ for r ${\geq}$ 0; p ${\geq}$ 0 with (1 + 2r)q ${\geq}$ p + 2r, Proc. Amer. Math. Soc., 101(1987), 85-87.
  14. F. Hansen, An operator inequality, Math. Ann., 246(1980), 249-250. https://doi.org/10.1007/BF01371046
  15. O. G¨uler, Barrier functions in interior point methods, Mathematics of Operations Research, 21(1996), 860-885. https://doi.org/10.1287/moor.21.4.860
  16. M. Koecher, Jordan algebras and their applications, Lecture Notes, University of Minnesota, 1962.
  17. A. Koranyi, Monotone functions on formally real Jordan algebras, Math. Ann., 269(1984), 73-76. https://doi.org/10.1007/BF01455996
  18. F. Kubo and T. Ando, Means of positive linear operators, Math. Ann., 246(1980), 205-224. https://doi.org/10.1007/BF01371042
  19. S. Lang, Fundamentals of differential geometry, Graduate Texts in Math., Springer 1999.
  20. J. D. Lawson and Y. Lim, The geometric mean, matrices, metrics, and more, Amer. Math. Monthly, 108(2001), 797-812. https://doi.org/10.2307/2695553
  21. J. D. Lawson and Y. Lim, Symmetric sets with midpoints and algebraically equivalent theories, Result. Math., 46(2004), 37-56. https://doi.org/10.1007/BF03322869
  22. J. D. Lawson and Y. Lim, Means on dyadic symmetric sets and polar decompositions, Abh. Math. Sem. Univ. Hamburg, 74(2004), 135-150. https://doi.org/10.1007/BF02941530
  23. J. D. Lawson and Y. Lim, Geometric means and reflection quasigroups, Quasigroups and Related Systems, 14(2006), 43-59.
  24. J. D. Lawson and Y. Lim, Symmetric spaces with convex metrics, Forum Math., to appear.
  25. J. D. Lawson and Y. Lim, Metric convexity of symmetric cones, submitted.
  26. Y. Lim, Finsler metrics on symmetric cones, Math. Ann., 316(2000), 379-389. https://doi.org/10.1007/s002080050017
  27. Y. Lim, Geometric means on symmetric cones, Arch. der Math., 75(2000), 39-45. https://doi.org/10.1007/s000130050471
  28. Y. Lim, Applications of geometric means on symmetric cones, Math. Ann., 319(2001), 457-468. https://doi.org/10.1007/PL00004442
  29. Y. Lim, Fixed points of monotone symplectic mappings, Math. Ann., 321(2001), 601-613. https://doi.org/10.1007/s002080100241
  30. Y. Lim, Best approximation in Riemannian geodesic submanifolds of positive definite matrices, Canad. J. Math., 56(2004), 776-793. https://doi.org/10.4153/CJM-2004-035-5
  31. K. -H. Neeb, A Cartan-Hadamard theorem for Banach-Finsler manifolds, Geometriae Dedicata, 95(2002), 115-156. https://doi.org/10.1023/A:1021221029301
  32. Y. E. Nesterov and M. J. Todd, Self-scaled barriers and interior-point methods for convex programming, Mathematics of Operations Research, 22(1997), 1-42. https://doi.org/10.1287/moor.22.1.1
  33. R. D. Nussbaum and J. E. Cohen, The arithmetic-geometric mean and its generalizations for noncommuting linear operators, Ann. Scuola. Norm. Sup. Pisa Cl. Sci. (4), 15(1988), 239-308.
  34. R. D. Nussbaum, Hilbert's projective metric and iterated nonlinear maps, Memoirs of AMS, 21(1988).
  35. R. D. Nussbaum, Finsler structures for the part metric and Hilbert's projective metric and applications to ordinary differential equations, Differential and Intergral Equations, 7(1994), 1649-1707.
  36. G. K. Pedersen and M. Takesaki, The operator equation THT = K; Proc. Amer. Math. Soc., 36(1972), 311-312.
  37. W. Pusz and S. L. Woronowicz, Functional calculus for sesquilinear forms and the purification map, Rep. Math. Phys., 8(1975), 159-170. https://doi.org/10.1016/0034-4877(75)90061-0
  38. O. S. Rothaus, Domains of positivity, Abh. Math. Sem. Univ. Hamburg, 24(1960), 189-235. https://doi.org/10.1007/BF02942030
  39. G. Trapp, The Ricatti equation and geometric mean, Contemporary Math., 47(1985), 437-445. https://doi.org/10.1090/conm/047/828317
  40. A. Unterberger and H. Upmeier, Pseudodifferential analysis on symmetric cones, Studies in Advanced Mathematics 1996.
  41. H. Upmeier, Symmetric Banach manifolds and Jordan C*-algebras, North Holland Mathematical Studies, 1985.